Complain, kvetch, whine, moan. Most often used in gerund form, i.e., whingeing. Interestingly, in U.S., spelling is frequently given as whing and whinging, which, sensibly (as the first g is soft, as in fringe), have not been common in U.K. in recent decades.
… money alone won’t do it [produce happiness]. Listen to the poor rich lottery winners whingeing away. (The word whinge, as used in my poker game, is a whine from a winner.) (Walter Goodman, New York Times, April 15, 1996)/And as much as I adore Sheldon’s persnickety nature, watching him devolve into a whingeing man-child, bitching about his mother not making him fried chicken or pecan pie, kept what had the potential to be a top-flight episode from ever taking off. (Entertainment Weekly, “Big Bang Theory Recap,” October 20, 2011)
notoneoffbritishisms.com 
We picked it up from the Australians, who used to call English incomers “whingeing Poms”. Before that we just used to whine.
Oh you should hear the Aussies when any of the Four Nations is beating them at rugby! In 2006, there was one ‘jokey’ piece about “we’ve hated the pommies ever since they came over and started killing our Aborigines”.
No, mate, we only gave you passage. What you did when you got there was your own affair.
~alec
OED actually dates it from 1150 (!) and says it originated in “Scottish and Northern dialect.” Joyce used it in 1922 in “Ulysses”: “You crossed her last wish in death and yet you sulk with me because I don’t whinge like some hired mute from Lalouette’s.”
I think when the Aussies use the term POM is it actually POHM, for Prisoner Of Her Majesty.
That’s held to be apocryphal, like SOS being “save our souls”. Some form of pomegranate is most likely.
~alec
“Pomegranate”? Oh, yes, of course! (??????)
I heard it came from the French word for apple (pomme). Who knows?
I totally & completely hate ‘whinge’.
It’s just way too cutesy!